Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes?

I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's some division going on and there are some fancy 'd' characters that are added in... I recall thinking that it was a lot of characters to represent an action with respect to one variable. Of course, once you start moving the dx around it makes a little more sense as to why they exist - but is this the only reason?

Any history lesson or examples where this notation is helpful or unhelpful is appreciated.

now, take the limit as ∆x goes to zero, and you have the differential. This is indicated by using a lower case d instead of the ∆.

df(x) / dx

Now, if this operation is treated as an operator applied to a function, it is usually represented as

d / dx (applied to f(x) )

Note that (typically in physics), you can also use the letter δ to indicate very small intervals and in general you would use the symbol ∂ to represent partial differentials. They are all variations of the letter D.

If you're a physics kind of person, then a good reason to like this notation is that it gives the correct units for the derivative: whatever units $f(x)$ is in, the units for $\frac{d}{dx} f(x)$ are obtained by dividing by the units for $x$.

This is the Leibniz notation, which is based on the ratio of "infinitesimals". dy and dx are, respectively, the infinitesimal increment of the dependent variable y and the infinitesimal increment of the variable x.

There are other notations: Newton notation, which puts a dot over the variable name, and Cauchy notation, which uses the operator D, as in D(sin(x))=cos(x).

In other words, in Leibniz notation, the use of the division notation is intentional.
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Heath HunnicuttJul 28 '10 at 20:11

2

Yes, right. One could give a rigorous definition of dx in term of differentials, if needed. The advantage of this notation is that some formulas become "obvious"; for example, the chain rule is $$\frac{df(g(x))}{dx}=\frac{df}{dg}\cdot\frac{dg}{dx}$$.
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zarAug 14 '10 at 20:24

Calculus used to be done with infinitesimals (Archimedes The Method of Mechanical Theorems, Newtons Fluxions, ...) but there was some controversy about these ghostly quantities eventually the whole foundation of analysis was rebuilt using limits but the old notations have been kept. So there is (as you noticed) a strange gap between reality (epsilonics) and intuition (infinitesimal quantities) but there are a few more recent redevelopments of the foundation of analysis for example Keisler or Bell.

It is possible to attach a precise meaning to "dx" as an infinitesimal using nonstandard analysis. There using model theory it is possible to rigorously make intuitive statements in calculus that we use all the time. This is a theory created by Abraham Robinson.

For a brief informal discussion, see the appendix(by Stewart) of "What is mathematics" by Courant, Robbins and Stewart. Or see Robinson's book "Nonstandard Analysis".

This page, (which quotes from "Yearning for the Impossible, The Surprising Truths of Mathematics") has:

...It has to be explained that dy/dx is not the ratio of infinitesimal differences dy and dx -- since infinitesimals do not exist -- but is rather a symbol for the limit of the ratio y/x as x tends to zero...

That doesn't seem to come as quite right in light of nonstandard analysis. Both dy and dx come as infinitesimals (though they need work out that way). E. G., (dy/dx)(x^2)=((x+i)^2-x^2)/i where i represents some infinitesimal. So we have (dy/dx)(x^2)=(2xi+i^2)/i. The denominator "i" comes as an infinitesimal by definition. The square of an infinitesimal, and the product of a real (or finite number) and an infinitesimal comes as an infinitesimal (see Keisler's book in anon's statement). The sum of two infinitesimals yields an infinitesimal, so dy also qualifies as an infinitesimal.
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Doug SpoonwoodJun 13 '11 at 21:33